Magnetic deformation theory of a vesicle

We have extended the Helfrich's spontaneous curvature model [M. Iwamoto and Z. C. Ou-Yang. Chem. Phys. Lett. 590(2013)183; Y. X. Deng, et al., EPL. 123(2018)68002] of the equilibrium vesicle shapes by adding the interaction between magnetic field and the constituent molecules to explain the phenomena of the reversibly deformation of artificial stomatocyte [P. G. van Rhee, et al., Nat. Commun. Sep 24;5:5010(2014), doi: 10.1038/ncomms6010] and the anharmonic deformation of a self-assembled nanocapsules of bola-amphiphilic molecules and the linear birefrin-gence [O.V. Manyuhina, et al., Phys. Rev. Lett. 98(2007)146101]. However, the so-phisticated mathematics in differential geometry is still covered. Here, we present the derivations of formulas in detailed to reveal the perturbation of deformation psi under two cases. New features such as the influence of temperature on the bend modulus of vesicle membrane have been revealed.